Abstract: Standard vectoring protocols, such as EIGRP, BGP, DSDV, or Babel, only route on optimal paths when the total order on path attributes that substantiates optimality is consistent with the extension operation that calculates path attributes from link attributes, leaving out many optimality criteria of practical interest. We present a solution to this problem and, more generally, to the problem of routing on multiple optimality criteria. A key idea is the derivation of a partial order on path attributes that is consistent with the extension operation and respects every optimality criterion of a designated collection of such criteria. We design new vectoring protocols that compute on partial orders, with every node capable of electing multiple attributes per destination rather than a single attribute as in standard vectoring protocols. Our evaluation over publicly available network topologies and attributes shows that the proposed protocols converge fast and enable optimal path routing concurrently for many optimality criteria with only a few elected attributes at each node per destination. We further show how predicating computations on partial orders allows incorporation of service chain constraints on optimal path routing.